Optimal Controller Design for Plants with Polynomial Nonlinearities

The paper deals with the development of theoretical backgrounds to design optimal control systems for nonlinear plants with quadratic nonlinearity in right-hand expression. These backgrounds are based on solving dynamic programming problem for one class of nonlinear plants. We use ordinary differential equations in the normal matrix form to define the dynamic of the considered plant. Then we apply feedback linearization transformation to rewrite the abovementioned equation as differential equations in the canonical form. The main advantage of such representation is the linear differential dependence between all state variables and only the last equation has a nonlinear function in this case. That is why we use the differential equations in the canonical form to define the perturbed motions equations and then the quadratic Lyapunov's function is defined as Newton's binomial. In this case Lyapunov's function is invariant to plant structure and parameters. The factors of this function are defined as factors of Newton's binomial. We use the defined in such a way Lyapunov function to solve dynamical programming problem and define control law as the Bellman's equation solution. The benefits of our approach are shown by considering an example of controller design for an electric drive with ventilator load. The numerical simulation shows that the designed system is asymptotically stable and has zero error.